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RADIATION  OF  HEAT ON  THE  PROPAGATION  OF  SOUND.     147
equilibrium, the cold has diminished in consequence of the supply of heat from the sides of the cylinder, and therefore the force urging the piston forward, arising, as it does, from the excess of the external over the internal pressure, is less than that which opposed the piston in moving from its position of equilibrium. Hence in this case the motion of the piston could not be kept up without a continual supply of labouring force. Lastly, suppose the piston to oscillate with great rapidity, so that there is not time for any sensible quantity of heat to pass and repass between the air and the sides of the cylinder. In this case the pressures would be equal when the piston was at a given point of the cylinder, whether it were going or returning, and consequently there would be no permanent consumption of labouring force. I do not speak of the disturbance of the external air, because I am not now taking into account the inertia of the air either within or without the cylinder. The third case, then, is similar to the first, so far as regards the permanence of the motion ; but there is this difference; that, in consequence of the heat produced by compression and the cold produced by rarefaction, the force urging the piston towards its position of equilibrium, on whichever side of that position the piston may happen to be, is greater than it would have been had the temperature remained unaltered.
Now the first case is analogous to that of the sonorous vibrations of air when the heat and cold produced by sudden condensation and rarefaction are supposed to pass away with great rapidity. For we are evidently concerned only with the relative rates at which the phase of vibration changes, and the heat causing the excess of temperature 9 passes away, so that it is perfectly immaterial whether we suppose the change of motion to be very slow, or the cooling of heated air to be very rapid. The second case is analogous to that of sound, when we suppose the constants q arid n comparable with each other; and we thus see how it is, that, on such a supposition, labouring force would be so rapidly consumed, and the sound so rapidly stifled. The third case is analogous to that of sound when we make the usual supposition, that the alternations of condensation and rarefaction take place with too great rapidity to allow a given portion of air to acquire or lose any sensible portion of heat by radiation. The increase in the force of restitution of the piston, arising from the alternate elevation and
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